operations on ideals in a subring of number field For three ideals $I, J$ and $K$ of a subring $R$ in a number field $L$,
does this equality hold in general?
$(I+J) \cap K = (I \cap K) + (J \cap K)$
I have no counterexample yet but I couldn't prove this.
Does this hold if $R$ is a finite local ring over $\mathbb{Z}_p$? 
(In other words, $R$ is a local ring and $R$ is a $\mathbb{Z}_p$ module of finite rank.)
 A: For the first question: distributivity of the lattice of ideals for a commutative (noetherian) ring with no zero divisors is equivalent to the ring being Pr\"ufer (Dedekind). [See the references in wikipedia]. So, for a counter-example which is not even integrally closed, in a quadratic extension of the rational numbers take the "naive integers": integer linear combinations of 1 and a square root of an integer in cases where to obtain the integral closure one needs linear combinations with half-integer coefficients (for example, cubic roots of 1).
For the second question, I still have to check Bourbaki's exercises for potential counter-examples. What is sure is that (1) for a noetherian local commutative ring being uniserial is the same as having distributive lattice of ideals, hence it is also equivalent to being a discrete valuation domains or artinian uniserial i.e. proper homomorphic image of a discrete valuation domain (see the chapters by Tuganbaev in the Handbook of algebra for much more general results); (2)  finitely generated modules over a noetherian uniserial ring (or more generally finitely presented modules over a serial ring) are (finite) direct sum of uniserial (finitely presented) submodules (Warfield's theorem). But when one applies this to a ring $R$ as module over a discrete valuation domain one does not obtain that $R$ is serial as module over itself (only over the valuation domain). 
I suspect that even finite rings $R$ whose additive group is a $p$-group could give counterexamples (as modules over the integers modulo $p^n$, and then a Hensel lemma technology should give a example over the $p$-adic integers), Or one could try the localizations (and then completions) of the number field examples above.
